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In the Torah, priests are called kohanim (singular: kohen), and are defined as the direct male descendants of Aharon HaKohen. There are many laws concerning kohanim specifically
that do not apply to other Jews, such as not marrying a divorcee or not becoming ritually impure. The Kohanim conducted all the services that were required to be done in the Temple.

A Rabbi, historically speaking, is someone who was ordained as such, based on his demonstrating an expertise in Jewish law. There is definitely no requirement for a rabbi to be a kohen, nor for a kohen to be a rabbi. Nowadays, the title "Rabbi" is more ambiguous. It can refer to anyone who is in a position of teaching Torah to others.

In fact, if someone who is a kohen applies for a rabbi job today (as spiritual leader for a congregation), the fact that he's a kohen can be a downside, as he can't officiate at funerals.
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ShalomMay 25 '11 at 13:22

A kohen serving in the Temple was expected to have [at least some of] the expertise in practical laws that is expected of a rabbi in modern times. In that sense, the two occupations overlap, although there are significant differences, covered in other answers.

The quotation below (with my English approximation) is one of the Torah's commands to kohanim to not only grasp the details of the law intellectually, but to teach those laws to the nation.

God spoke to Aharon saying, "do not drink wine or [some other kind of] wine - you or your sons with you - when you enter the temple. And [thus] do not come to die. [This is] an eternal statute for your progeny. And [so] to differentiate between the sanctified and the mundane, and between the pure and the impure, and to teach Israel all of the statutes which God has spoken to them by way of Moshe."

cohen is simple the person that works at the temple. while restritions apply, to qualify to this job you basically need that your father worked there too. it's a kind of proteccionism.
a rabbi is basically someone that studied the necessary to qualify, the qualification is based on merit.
you could have a rabbi that is also a cohen and you could have a cohen one thing doesn't affect the other, it's like your father puts you to work on the family business, no matter if you have a diploma on administration. you could have the diploma or not.
while someone even with a doctorate wouldn't get the job, because your father only puts you, your brothers and cousins to work there.